Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, D

Average Error: 1.2 → 0.3

Time: 6.9s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]

↓

\[\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\right)\]

C

double f(double x, double y, double z, double t) {
        double r733054 = 1.0;
        double r733055 = 3.0;
        double r733056 = r733054 / r733055;
        double r733057 = x;
        double r733058 = y;
        double r733059 = 27.0;
        double r733060 = r733058 * r733059;
        double r733061 = r733057 / r733060;
        double r733062 = r733055 * r733061;
        double r733063 = z;
        double r733064 = 2.0;
        double r733065 = r733063 * r733064;
        double r733066 = r733062 / r733065;
        double r733067 = t;
        double r733068 = sqrt(r733067);
        double r733069 = r733066 * r733068;
        double r733070 = acos(r733069);
        double r733071 = r733056 * r733070;
        return r733071;
}

↓

double f(double x, double y, double z, double t) {
        double r733072 = 1.0;
        double r733073 = 3.0;
        double r733074 = cbrt(r733073);
        double r733075 = r733074 * r733074;
        double r733076 = r733072 / r733075;
        double r733077 = 1.0;
        double r733078 = r733077 / r733074;
        double r733079 = x;
        double r733080 = y;
        double r733081 = 27.0;
        double r733082 = r733080 * r733081;
        double r733083 = r733079 / r733082;
        double r733084 = r733073 * r733083;
        double r733085 = z;
        double r733086 = 2.0;
        double r733087 = r733085 * r733086;
        double r733088 = r733084 / r733087;
        double r733089 = t;
        double r733090 = sqrt(r733089);
        double r733091 = r733088 * r733090;
        double r733092 = acos(r733091);
        double r733093 = r733078 * r733092;
        double r733094 = r733076 * r733093;
        return r733094;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	1.2
Target	1.2
Herbie	0.3

\[\frac{\cos^{-1} \left(\frac{\frac{x}{27}}{y \cdot z} \cdot \frac{\sqrt{t}}{\frac{2}{3}}\right)}{3}\]

Derivation

1. Initial program 1.2

   \[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
2. Using strategy rm

3. Applied add-cube-cbrt 1.2

   \[\leadsto \frac{1}{\color{blue}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}}} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]

4. Applied *-un-lft-identity 1.2

   \[\leadsto \frac{\color{blue}{1 \cdot 1}}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]

5. Applied times-frac 0.3

   \[\leadsto \color{blue}{\left(\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{1}{\sqrt[3]{3}}\right)} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]

6. Applied associate-*l* 0.3

   \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\right)}\]

7. Final simplification 0.3

   \[\leadsto \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\right)\]

Reproduce

herbie shell --seed 2020045 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, D"
  :precision binary64

  :herbie-target
  (/ (acos (* (/ (/ x 27) (* y z)) (/ (sqrt t) (/ 2 3)))) 3)

  (* (/ 1 3) (acos (* (/ (* 3 (/ x (* y 27))) (* z 2)) (sqrt t)))))